Abstract
This chapter covers extensions of the basic Description Logics introduced in Chapter 2 by very expressive constructs that require advanced reasoning techniques. In particular, we study reasoning in Description Logics that include general inclusion axioms, inverse roles, number restrictions, reflexive-transitive closure of roles, fixpoint constructs for recursive definitions, and relations of arbitrary arity. The chapter will also address reasoning w.r.t. knowledge bases including both a TBox and an ABox, and discuss more general ways to treat objects. Since the logics considered in the chapter lack the finite model property, finite model reasoning is of interest and will also be discussed. Finally, we mention several extensions to Description Logics that lead to undecidability, confirming that the expressive Description Logics considered in this chapter are close to the boundary between decidability and undecidability.